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Prevention Although the risk of Legionnaires' disease being spread by large-scale water systems cannot be eliminated, it can be greatly reduced by writing and enforcing a highly detailed, systematic water safety plan appropriate for the specific facility involved (office building, hospital, hotel, spa, cruise ship, et...
Legionnaires' disease
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Biology and health sciences
Bacterial infections
Health
Treatment Effective antibiotics include most macrolides, tetracyclines, ketolides, and quinolones. Legionella spp. multiply within the cell, so any effective treatment must have excellent intracellular penetration. Current treatments of choice are the respiratory tract quinolones (levofloxacin, moxifloxacin, gemifloxac...
Legionnaires' disease
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Bacterial infections
Health
Prognosis The fatality rate of Legionnaires' disease has ranged from 5–30% during various outbreaks and approaches 50% for nosocomial infections, especially when treatment with antibiotics is delayed. Hospital-acquired Legionella pneumonia has a fatality rate of 28%, and the principal source of infection in such cases ...
Legionnaires' disease
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Health
A 2011 study successfully used modeling to predict the likely number of cases during Legionnaires' outbreaks based on symptom onset dates from past outbreaks. In this way, the eventual likely size of an outbreak can be predicted, enabling efficient and effective use of public-health resources in managing an outbreak. ...
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In April 1985, 175 people in Stafford, England, were admitted to the District or Kingsmead Stafford Hospitals with chest infection or pneumonia. A total of 28 people died. Medical diagnosis showed that Legionnaires' disease was responsible and the immediate epidemiological investigation traced the source of the infecti...
Legionnaires' disease
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Twelve people were diagnosed with the disease in an outbreak in the Bronx, New York, in December 2014; the source was traced to contaminated cooling towers at a housing development. In July and August 2015, another, unrelated outbreak in the Bronx killed 12 people and made about 120 people sick; the cases arose from a ...
Legionnaires' disease
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In December 2019, the government of Western Australia's Department of Health was notified of four cases of Legionnaires' disease. Those exposed had recently visited near Bali's Ramayana Resort and Spa in central Kuta. In February 2024, Minnesota Department of Health issued a news release stating that fourteen (14) cas...
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The Rho Ophiuchi cloud complex is a complex of interstellar clouds with different nebulae, particularly a dark nebula which is centered 1° south of the star ρ Ophiuchi, which it among others extends to, of the constellation Ophiuchus. At an estimated distance of about , or 460 light years, it is one of the closest sta...
Rho Ophiuchi cloud complex
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Notable nebulae
Astronomy
The first brown dwarf to be identified in a star-forming region was Rho Oph J162349.8-242601, located in the Rho Ophiuchi cloud. One of the older objects at the edge of the primary star-forming region was found to be a circumstellar disk seen nearly edge-on. It spans a diameter of 300 AU and contains at least twice the...
Rho Ophiuchi cloud complex
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Astronomy
In computing and computer science, a processor or processing unit is an electrical component (digital circuit) that performs operations on an external data source, usually memory or some other data stream. It typically takes the form of a microprocessor, which can be implemented on a single or a few tightly integrated ...
Processor (computing)
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There are several forms of processors specialized for machine learning. These fall under the category of AI accelerators (also known as neural processing units, or NPUs) and include vision processing units (VPUs) and Google's Tensor Processing Unit (TPU). Sound chips and sound cards are used for generating and process...
Processor (computing)
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The Delhi Metro is a rapid transit system that serves Delhi and the adjoining satellite cities of Ghaziabad, Faridabad, Gurugram, Noida, Bahadurgarh, and Ballabhgarh in the National Capital Region of India. The system consists of 10 colour-coded lines serving 257 stations, with a total length of . It is India's largest...
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The city expanded significantly while technical studies and financing the project underway, doubling its population and increasing the number of vehicles five-fold between 1981 and 1998. Traffic congestion and pollution soared as an increasing number of commuters used private vehicles, and the existing bus system was u...
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The Delhi Metro's first line, the Red Line, was inaugurated by Prime Minister Atal Bihari Vajpayee on 24 December 2002. The metro became India's second underground rapid transit system, after the Kolkata Metro, when the Vishwa Vidyalaya–Kashmere Gate section of the Yellow Line opened on 20 December 2004. The undergroun...
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Work on Phase III began in 2011, with 2016 the planned deadline. Over 20 tunnel-boring machines were used simultaneously to expedite construction, which was completed in March 2019 (except for a small stretch due to non-availability of land). Short extensions were later added to Phase III, which was expected to be comp...
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Construction Incidents On 19 October 2008, a launching gantry and part of the overhead Blue Line extension under construction in Laxmi Nagar collapsed and fell on a passing bus. Workers were using a crane to lift a 400-tonne concrete span of the bridge when the gantry and a span of the bridge collapsed on the bus. The...
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Gurugram Metro loop (from HUDA City Centre to Cyber City) - approved: The total length of the corridor will be about , consisting of 27 elevated stations with six interchange stations. This link would start at HUDA City Centre and move towards Sector 45, Cyber Park, district shopping centre, Sector 47, Subhash Chowk, S...
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Gurgaon – Faridabad metro - DPR ready: In May 2020, the Detailed Project Report (DPR) and survey for the long Gurgaon-Faridabad metro link from Vatika Chowk in Gururam to Bata Chowk in Faridabad was completed which will have 8 stations, of which the elevated stretch along the Gurgaon-Faridabad Road through eco-sensit...
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Uttar Pradesh (UP) projects Shiv Vihar – Loni: Proposed but not approved Noida – Noida International Airport: surface line along the Yamuna Expressway serving the proposed Noida International Airport. The line, envisioned to be completed by 2025, will connect with the Noida Metro. Integration with RapidX The Rapid...
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The Red Line, the first metro line opened, connects Rithala in the west to Shaheed Sthal (New Bus Adda) in the east for a distance of . Partly elevated and partly at grade, it crosses the Yamuna River between the Kashmere Gate and Shastri Park stations. The opening of the first stretch on 24 December 2002, between Shah...
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An additional stretch from Qutab Minar to Millennium City Centre Gurugram, initially operating separately from the mainline, opened on 21 June 2010; the Chhatarpur station on this stretch opened on 26 August of that year. Due to delays in acquiring land to construct the station, it was built with prefabricated structur...
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A branch of the Blue Line, inaugurated on 8 January 2010, runs for from the Yamuna Bank station to Anand Vihar in East Delhi. It was extended to Vaishali on 14 July 2011. A stretch from Dwarka Sector 9 to Dwarka Sector 21 opened on 30 October 2010. On 9 March 2019, a extension from Noida City Centre to Noida Electro...
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It opened in two stages, with the Inderlok–Mundka section opening on 3 April 2010 and the Kirti Nagar–Ashok Park Main branch line opening on 27 August 2011. On 6 August 2012, to improve commuting in the National Capital Region, the government of India approved an extension from Mundka to Bahadurgarh in Haryana. The ...
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The line's Faridabad corridor is the longest corridor outside Delhi: 11 stations and . On 28 May 2017, the ITO–Kashmere Gate corridor was opened by Union Minister of Urban Development Venkaiah Naidu and Chief Minister of Delhi Arvind Kejriwal. The underground section is popularly known as the Heritage Line. Interchange...
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Originally scheduled to open before the 2010 Commonwealth Games, the line failed to obtain the mandatory safety clearance and was opened on 23 February 2011 after a delay of about five months. Sixteen months after beginning operations, it was shut down for viaduct repairs on 7 July 2012. The line reopened on 22 January...
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The Pink Line has 38 stations from Majlis Park to Shiv Vihar, both in North Delhi. With a length of , it is the Delhi Metro's longest line. The mostly-elevated line covers Delhi in a U-shaped pattern. It is also known as the Ring Road Line, since it runs along the busy Ring Road. The line has interchanges with most of...
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The Grey Line (also known as Line 9), the metro's shortest, runs from Dwarka to Dhansa Bus Stand in western Delhi. The line has four stations (Dhansa Bus Stand, Najafgarh, Nangli and Dwarka), and has an interchange with the Blue Line at Dwarka. The Najafgarh-to-Dwarka section was opened on 4 October 2019. The extensio...
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Trains operate at a frequency of one to two minutes to five to ten minutes between 05:00 and 00:00, depending upon peak and off-peak hours. They typically travel up to , and stop for about 20 seconds at each station. Automated station announcements are in Hindi and English. Many stations have ATMs, food outlets, cafés,...
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The metro's fares were last revised on 10 October 2017, based on the recommendation of the 4th Fare Fixation Committee in May 2016. Metro commuters have five choices for ticket purchases: RFID token: RFID tokens are valid only for a single journey on the day of purchase. Their value depends on the distance travelled, ...
Delhi Metro
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As the metro has expanded, high ridership on new trains has led to increasing overcrowding and delays. To alleviate the problem, eight-coach trains have been introduced on the Yellow and Blue Lines and more-frequent trains have been proposed. Infrequent, overcrowded and erratic feeder bus services connecting stations t...
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^ From 2019 onwards the DMRC changed the ridership calculation to count every trip taken by a passenger on a line. This means that a passenger that takes 2 connections will count 3 times towards ridership. This is different from the more standard practice of counting entire journeys applied in other metro systems. Fin...
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The metro has two rail gauges. Phase I lines have broad gauge rolling stock, and three Phase II lines have rolling stock. Trains are maintained at seven depots at Khyber Pass and Sultanpur for the Yellow Line, Mundka for the Green Line, Najafgarh and Yamuna Bank for the Blue Line, Shastri Park for the Red Line, and S...
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Phase II rolling stock is supplied by Bombardier Transportation, which received an order for 614 cars at a cost of about . Although the initial trains were made in Görlitz, Germany and Sweden, the remainder will be built at Bombardier's factory in Savli (near Vadodara). The four- and six-car trains have a capacity of 1...
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The Airport Express line introduced WiFi service at all its stations on 13 January 2012. Connectivity in trains is expected in the future. WiFi service is provided by YOU Broadband and Cable India. In August 2017, Wifi service began at all the 50 stations of the Blue Line. A fully-automated, operator-less train system ...
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The metro has been promoted as an integral part of community infrastructure, and artwork depicting the local way of life has been displayed at stations. Students at local art colleges have designed murals at metro stations, and the viaduct pillars of some elevated sections have been decorated with mosaic murals created...
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The Abyssinian is a breed of cat with a distinctive "ticked" tabby coat, in which individual hairs are banded with different colours. They are also known simply as Abys. The first members of the breed to be exhibited in England were brought there from Abyssinia (now known as Ethiopia), hence the name. Genetic studies...
Abyssinian cat
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Biology and health sciences
Cats
Animals
The Abyssinian is one of the oldest established cat breeds, being recognised in 1929 by the Governing Council of the Cat Fancy. The breed was developed in the United Kingdom with references dating back to the 1890s. Description Appearance The Abyssinian is a lithe, fine-boned, muscular, medium to large sized cat. The...
Abyssinian cat
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Biology and health sciences
Cats
Animals
The breed's original colour standard is a warm deep reddish-brown base with black ticking, known as "usual" in the United Kingdom, "tawny" in Australia, and "ruddy" elsewhere. Sorrel (also called cinnamon or red), a lighter coppery base with chocolate brown ticking, is a unique mutation of this original pattern. Other ...
Abyssinian cat
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Biology and health sciences
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Animals
Health Familial renal amyloidosis or AA amyloidosis, a kidney disorder due to a mutation in the AA amyloid protein gene, has been seen in Abyssinians. The Abyssinian has had severe problems with blindness caused by a hereditary retinal degeneration due to mutations in the rdAc gene. However, the prevalence has been red...
Abyssinian cat
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Biology and health sciences
Cats
Animals
The Abyssinian is the cat breed most commonly affected by progressive retinal atrophy. The condition is caused by two separate mutations in the breed. Early onset PRA is caused by an autosomal dominant mutation in the CRX gene. Late onset PRA is caused by an autosomal recessive mutation in the CEP290 gene. The Abyssin...
Abyssinian cat
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Biology and health sciences
Cats
Animals
Plant breeding is the science of changing the traits of plants in order to produce desired characteristics. It is used to improve the quality of plant products for use by humans and animals. The goals of plant breeding are to produce crop varieties that boast unique and superior traits for a variety of applications. Th...
Plant breeding
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Gregor Mendel (1822–84) is considered the "father of genetics". His experiments with plant hybridization led to his establishing laws of inheritance. Genetics stimulated research to improve crop production through plant breeding. Selective breeding played a crucial role in the Green Revolution of the 20th century. M...
Plant breeding
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Gartons Agricultural Plant Breeders in England was established in 1880, which became a public company in 1898, by John Garton, who was one of the first to commercialize new varieties of agricultural crops created through cross-pollination. The firm's first introduction was the , an oat variety. It is one of the first a...
Plant breeding
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When distantly related species are crossed, plant breeders make use of a number of plant tissue culture techniques to produce progeny from otherwise fruitless mating. Interspecific and intergeneric hybrids are produced from a cross of related species or genera that do not normally sexually reproduce with each other. Th...
Plant breeding
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When a desirable trait has been bred into a species, a number of crosses to the favored parent are made to make the new plant as similar to the favored parent as possible. Returning to the example of the mildew resistant pea being crossed with a high-yielding but susceptible pea, to make the mildew resistant progeny of...
Plant breeding
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Doubled haploidy and reverse breeding Homozygous plants with desirable traits can be produced from heterozygous starting plants, if a haploid cell with the alleles for those traits can be produced, and then used to make a doubled haploid. The doubled haploid will be homozygous for the desired traits. Furthermore, two...
Plant breeding
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To genetically modify a plant, a genetic construct must be designed so that the gene to be added or removed will be expressed by the plant. To do this, a promoter to drive transcription and a termination sequence to stop transcription of the new gene, and the gene or genes of interest must be introduced to the plant. A...
Plant breeding
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Genetic modification of plants that can produce pharmaceuticals (and industrial chemicals), sometimes called pharming, is a rather radical new area of plant breeding. The debate surrounding genetically modified food during the 1990s peaked in 1999 in terms of media coverage and risk perception, and continues today – f...
Plant breeding
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Genomic selection (GS) The NGS platform has substantially declined the time and cost required for sequencing and facilitated SNP discovery in model and non-model plants. This in turn has led to employing large-scale SNP markers in genomic selection approaches which aim at predicting genomic breeding values/GEBVs of ge...
Plant breeding
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Evolutionary breeding populations have been used to establish self-regulating plant–pathogen systems. Examples include barley, where breeders were able to improve resistance to Rynchosporium secalis scald over 45 generations. An evolutionary breeding project grew F5 hybrid bulk soybean populations on soil infested by t...
Plant breeding
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Conventional breeding intentionally limits phenotype plasticity within genotypes and limits variability between genotypes. Uniformity does not allow crops to adapt to climate change and other biotic stresses and abiotic stresses. Plant breeders' rights Plant breeders' rights is an important and controversial issue. P...
Plant breeding
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Maintaining specific conditions When new plant breeds or cultivars are bred, they must be maintained and propagated. Some plants are propagated by asexual means while others are propagated by seeds. Seed propagated cultivars require specific control over seed source and production procedures to maintain the integrity...
Plant breeding
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Yield With an increasing population, the production of food needs to increase with it. It is estimated that a 70% increase in food production is needed by 2050 in order to meet the Declaration of the World Summit on Food Security. But with the degradation of agricultural land, simply planting more crops is no longer ...
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There are many classical and modern breeding techniques that can be utilized for crop improvement in organic agriculture despite the ban on genetically modified organisms. For instance, controlled crosses between individuals allow desirable genetic variation to be recombined and transferred to seed progeny via natural...
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The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system: that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-metre (C⋅m). The debye (D) is another unit of measurement used in atomic physics and chemistry...
Electric dipole moment
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Physical sciences
Electrostatics
Physics
An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. The torque tends to align the dipole with the field. A dipole aligned parallel to an electric field has lower potential energy than a dipole making some non-zero angle with it. For a spatially uniform elect...
Electric dipole moment
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Physics
where r is the point of observation and di = ri − ri, ri being the position of the negative charge in the dipole i, and ri the position of the positive charge. This is the vector sum of the individual dipole moments of the neutral charge pairs. (Because of overall charge neutrality, the dipole moment is independent of ...
Electric dipole moment
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Physical sciences
Electrostatics
Physics
Thus, although two closely spaced opposite charges are not quite an ideal electric dipole (because their potential at short distances is not that of a dipole), at distances much larger than their separation, their dipole moment p appears directly in their potential and field. As the two charges are brought closer toge...
Electric dipole moment
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Physical sciences
Electrostatics
Physics
A formulation of Maxwell's equations based upon division of charges and currents into "free" and "bound" charges and currents leads to introduction of the D- and P-fields: where P is called the polarization density. In this formulation, the divergence of this equation yields: and as the divergence term in E is the to...
Electric dipole moment
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Physics
where the vector identity was used in the last steps. The first term can be transformed to an integral over the surface bounding the volume of integration, and contributes a surface charge density, discussed later. Putting this result back into the potential, and ignoring the surface charge for now: where the volume...
Electric dipole moment
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Physical sciences
Electrostatics
Physics
with dA0 an elementary area of the surface bounding the charges. In words, the potential due to a constant p inside the surface is equivalent to that of a surface charge which is positive for surface elements with a component in the direction of p and negative for surface elements pointed oppositely. (Usually the dire...
Electric dipole moment
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Physical sciences
Electrostatics
Physics
A physically more realistic modeling of p(r) would have the dipole moment density drop off rapidly, but smoothly to zero at the boundary of the confining region, rather than making a sudden step to zero density. Then the surface charge will not concentrate in an infinitely thin surface, but instead, being the divergenc...
Electric dipole moment
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Physical sciences
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Physics
General media If observation is confined to regions sufficiently remote from a system of charges, a multipole expansion of the exact polarization density can be made. By truncating this expansion (for example, retaining only the dipole terms, or only the dipole and quadrupole terms, or etc.), the results of the previo...
Electric dipole moment
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Physical sciences
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Physics
Electric dipole moments of fundamental particles Not to be confused with the magnetic dipole moments of particles, much experimental work is continuing on measuring the electric dipole moments (EDM; or anomalous electric dipole moment) of fundamental and composite particles, namely those of the electron and neutron, r...
Electric dipole moment
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Physics
where k is a constant and is the time correlation function of the total dipole moment. In general the total dipole moment have contributions coming from translations and rotations of the molecules in the sample, Therefore, the dielectric constant (and the conductivity) has contributions from both terms. This approach...
Electric dipole moment
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Pinguicula, commonly known as butterworts, is a genus of carnivorous flowering plants in the family Lentibulariaceae. They use sticky, glandular leaves to lure, trap, and digest insects in order to supplement the poor mineral nutrition they obtain from the environment. Of the roughly 80 currently known species, 13 are ...
Pinguicula
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Biology and health sciences
Lamiales
Plants
Many butterworts cycle between rosettes composed of carnivorous and non-carnivorous leaves as the seasons change, so these two ecological groupings can be further divided according to their ability to produce different leaves during their growing season. If the growth in the summer is different in size or shape to that...
Pinguicula
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Biology and health sciences
Lamiales
Plants
The leaf blade of a butterwort is smooth, rigid, and succulent, usually bright green or pinkish in colour. Depending on species, the leaves are between 2 and 30 cm (1-12") long. The leaf shape depends on the species, but is usually roughly obovate, spatulate, or linear. They can also appear yellow in color with a soft...
Pinguicula
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Lamiales
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The holes in the cuticle which allow for this digestive mechanism also pose a challenge for the plant, since they serve as breaks in the cuticle (waxy layer) that protects the plant from desiccation. As a result, most butterworts live in humid environments. Butterworts are usually only able to trap small insects and t...
Pinguicula
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Lamiales
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Fruit and seed The round to egg-shaped seed capsules open when dry into two halves, exposing numerous small (0.5–1 mm), brown seeds. If moisture is present the silique closes, protecting the seed and opening again upon dryness to allow for wind dispersal. Many species have a net-like pattern on their seed surface to a...
Pinguicula
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Lamiales
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Vegetative propagation As well as sexual reproduction by seed, many butterworts can reproduce asexually by vegetative reproduction. Many members of the genus form offshoots during or shortly after flowering (e.g., P. vulgaris), which grow into new genetically identical adults. A few other species form new offshoots us...
Pinguicula
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In general, butterworts grow in nutrient-poor, alkaline soils. Some species have adapted to other soil types, such as acidic peat bogs (ex. P. vulgaris, P. calyptrata, P. lusitanica), soils composed of pure gypsum (P. gypsicola and other Mexican species), or even vertical rock walls (P. ramosa, P. vallisneriifolia, and...
Pinguicula
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In 1583, Clusius already distinguished between two forms in his Historia stirpium rariorum per Pannoniam, Austriam: a blue-flowered form (P. vulgaris) and a white-flowered form (Pinguicula alpina). Linnaeus added P. villosa and P. lusitanica when he published his Species Plantarum in 1753. The number of known species r...
Pinguicula
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A detailed study of the phylogenetics of butterworts by Cieslak et al. (2005) found that all of the currently accepted subgenera and many of the sections were polyphyletic. The diagram below gives a more accurate representation of the correct cladogram. Polyphyletic sections are marked with an *. ┌─...
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A heart transplant, or a cardiac transplant, is a surgical transplant procedure performed on patients with end-stage heart failure or severe coronary artery disease when other medical or surgical treatments have failed. , the most common procedure is to take a functioning heart, with or without both lungs, from a recen...
Heart transplantation
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Not having a human donor heart available, James D. Hardy of the University of Mississippi Medical Center transplanted the heart of a chimpanzee into the chest of dying Boyd Rush in the early morning of Jan. 24, 1964. Hardy used a defibrillator to shock the heart to restart beating. This heart did beat in Rush's chest f...
Heart transplantation
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Norman Shumway performed the first adult heart transplant in the United States on January 6, 1968, at the Stanford University Hospital. A team led by Donald Ross performed the first heart transplant in the United Kingdom on May 3, 1968. These were allotransplants, the technical term for a transplant from a non-genetic...
Heart transplantation
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About 800,000 people have NYHA Class IV heart failure symptoms indicating advanced heart failure. The great disparity between the number of patients needing transplants and the number of procedures being performed spurred research into the transplantation of non-human hearts into humans after 1993. Xenografts from othe...
Heart transplantation
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Absolute contraindications: Irreversible kidney, lung, or liver disease Active cancer if it is likely to impact the survival of the patient Life-threatening diseases unrelated to the cause of heart failure, including acute infection or systemic disease such as systemic lupus erythematosus, sarcoidosis, or amyloidosi...
Heart transplantation
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A 2022 pilot study examining the acceptability and feasibility of using video directly observed therapy to increase medication adherence in adolescent heart transplant patients showed promising results of 90.1% medication adherence compared to 40-60% typically. Higher medication variability levels can lead to greater o...
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In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system represented by the state. Knowledge of the quantum state, and the rules for...
Quantum state
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Quantum mechanics
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The fundamentally statistical or probabilisitic nature of quantum measurements changes the role of quantum states in quantum mechanics compared to classical states in classical mechanics. In classical mechanics, the initial state of one or more bodies is measured; the state evolves according to the equations of motion;...
Quantum state
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In formal quantum mechanics (see below) the theory develops in terms of abstract 'vector space', avoiding any particular representation. This allows many elegant concepts of quantum mechanics to be expressed and to be applied even in cases where no classical analog exists. Wave function representations Wave functio...
Quantum state
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The postulates of quantum mechanics state that pure states, at a given time , correspond to vectors in a separable complex Hilbert space, while each measurable physical quantity (such as the energy or momentum of a particle) is associated with a mathematical operator called the observable. The operator serves as a lin...
Quantum state
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which involves superposition of joint spin states for two particles with spin 1/2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one ...
Quantum state
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Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state. More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses...
Quantum state
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Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with Dirac picture. Formalism in quantum physics Pure states a...
Quantum state
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As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the po...
Quantum state
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In general, the expression for probability always consist of a relation between the quantum state and a portion of the spectrum of the dynamical variable (i.e. random variable) being observed. For example, the situation above describes the discrete case as eigenvalues belong to the point spectrum. Likewise, the wave f...
Quantum state
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A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see Quantum statistical mechanics). Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fu...
Quantum state
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The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable is given by where and are eigenkets and eigenvalues, respectively, for the operator , and "" denotes tra...
Quantum state
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In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the...
Riemann sphere
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Complex analysis
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Rational functions Any rational function (in other words, is the ratio of polynomial functions and of with complex coefficients, such that and have no common factor) can be extended to a continuous function on the Riemann sphere. Specifically, if is a complex number such that the denominator is zero but the n...
Riemann sphere
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Topologically, the resulting space is the one-point compactification of a plane into the sphere. However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined complex structure, so that around every point on the sphere there is a neighborhood that can be biholomorphically identified...
Riemann sphere
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The transition maps between -coordinates and -coordinates are obtained by composing one projection with the inverse of the other. They turn out to be and , as described above. Thus the unit sphere is diffeomorphic to the Riemann sphere. Under this diffeomorphism, the unit circle in the -chart, the unit circle in the ...
Riemann sphere
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Conversely, let denote the sphere (as an abstract smooth or topological manifold). By the uniformization theorem there exists a unique complex structure on up to conformal equivalence. It follows that any metric on is conformally equivalent to the round metric. All such metrics determine the same conformal geometry....
Riemann sphere
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If one endows the Riemann sphere with the Fubini–Study metric, then not all Möbius transformations are isometries; for example, the dilations and translations are not. The isometries form a proper subgroup of , namely . This subgroup is isomorphic to the rotation group , which is the group of symmetries of the unit sph...
Riemann sphere
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Molecular cloning is a set of experimental methods in molecular biology that are used to assemble recombinant DNA molecules and to direct their replication within host organisms. The use of the word cloning refers to the fact that the method involves the replication of one molecule to produce a population of cells with...
Molecular cloning
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Biotechnology
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Virtually any DNA sequence can be cloned and amplified, but there are some factors that might limit the success of the process. Examples of the DNA sequences that are difficult to clone are inverted repeats, origins of replication, centromeres and telomeres. There is also a lower chance of success when inserting large-...
Molecular cloning
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Biotechnology
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In silico cloning and simulations Before actual cloning experiments are performed in the lab, most cloning experiments are planned in a computer, using specialized software. Although the detailed planning of the cloning can be done in any text editor, together with online utilities for e.g. PCR primer design, dedicate...
Molecular cloning
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